Tim, thank you so much for a speedy response. It is 2011 and you have dispelled the rumor that "this plugin has been abandoned since 2007".
I have installed the update you have released. However, the problem seems to be the same. No download counts or dashboad stats are appearing for already published podcsts. After I post this, I will attempt to publish a new podcast to see if I get the desired result.
To answer your question about the other tables in my database, all of them say "MyISAM" under the type column.
One thing I didn't mention in my previous post was that at one point I tried to uninstall podPress completely using the plugin uninstaller. I was hoping that when I installed it again it would recreate the database. This didn't work and in fact, when I reinstalled podPress, my previous podcasts magically came back! Leading me to believe, there is actually some database recording going on somewhere, despite the fact I cannot find it.
I have the phpmyadmin plugin installed. I would be happy to give you access to my admin area if you think it would help you diagnose the problem.